118th-octave temperaments: Difference between revisions

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[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets.

[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].

=== Parakleischis ===

== Parakleischis ==

118edo and its ~~multiples are members of both~~ [[parakleismic]] and [[Schismatic family|schismic]], and ~~from this it derives its name~~.

118edo's is an excellent 5-limit system and its comma basis constitutes the [[parakleismic]] and [[Schismatic family|schismic]] temperaments together. Parakleischis retains the 5-limit mapping from 118edo and leaves other harmonics as independent generators.

[[Subgroup]]: 2.3.5.7

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[[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]

~~Mapping~~ generators: ~15625/15552, ~7

: mapping generators: ~15625/15552, ~7

[[Optimal tuning]] ([[~~POTE~~]]): ~7/4 = 968.7235

[[Optimal tuning]] ([[CTE]]): ~7/4 = 968.7235

{{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}

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[[Badness]]: 0.145166

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

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Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]

Optimal tuning (~~POTE~~): ~7/4 = 968.5117

: mapping generators: ~176/176, ~7

Optimal tuning (CTE): ~7/4 = 968.5117

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Badness: 0.049316

===~~= Centenniamajor =~~===

=== Peithoian ===

~~Named after~~ the ~~fact that 18~~ is the ~~age of majority in most countries~~, and ~~100 (centennial) + 18 (major) =~~ 118.

Peithoian is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of peithoian temperament is to expand on that. Named after the minor planet [[wikipedia:118 Peitho|118 Peitho]].

Subgroup: 2.3.5.7.11

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Badness: 0.357

===== 13-limit =====

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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Badness: 0.122

=== Oganesson ===

== Oganesson ==

Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.

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[[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068

{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, ~~3442~~, 3776, 7198cd, 10974bccdd }}

{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3422, 3776, 7198cd, 10974bccdd }}

[[Badness]]: 2.66

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

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Optimal tuning (CTE): ~1953125/1354752 = 634.0085

{{Optimal ET sequence|legend=1| 354, 3068e, ~~3442~~, 3776, 11682ccdde }}

{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776, 11682ccdde }}

Badness: 0.568

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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Badness: 0.172

==== 17-limit ====

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

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~~Badness~~: 0.~~105~~

=== 19-limit ===

The closest superparticular to one step of 118edo is [[171/170]], so 19-limit extension for oganesson is prescribed.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 4096/4095, 6175/6174, 9801/9800, 14365/14364, 28900/28899, 3438981/3437500

Mapping: [{{val| 118 0 274 643 1094 499 607 1000}}, {{val| 0 3 0 -5 -11 -1 2 -8}}]

: mapping generators: ~171/170, ~238/165

Optimal tuning (CTE): ~238/165 = 634.006

[[Category:118edo]]

@@ Line 1: / Line 1: @@
-[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets.
+{{Infobox fractional-octave|118}}
+[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].
-=== Parakleischis ===
+== Parakleischis ==
-edo and its multiples are members of both [[parakleismic]] and [[Schismatic family|schismic]], and from this it derives its name.
+edo's is an excellent 5-limit system and its comma basis constitutes the [[parakleismic]] and [[Schismatic family|schismic]] temperaments together. Parakleischis retains the 5-limit mapping from 118edo and leaves other harmonics as independent generators.
 [[Subgroup]]: 2.3.5.7
@@ Line 10: / Line 11: @@
 [[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]
-Mapping generators: ~15625/15552, ~7
+: mapping generators: ~15625/15552, ~7
-[[Optimal tuning]] ([[POTE]]): ~7/4 = 968.7235
+[[Optimal tuning]] ([[CTE]]): ~7/4 = 968.7235
 {{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}
@@ Line 18: / Line 19: @@
 [[Badness]]: 0.145166
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 25: / Line 26: @@
 Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]
-Optimal tuning (POTE): ~7/4 = 968.5117
+: mapping generators: ~176/176, ~7
+Optimal tuning (CTE): ~7/4 = 968.5117
 {{Optimal ET sequence|legend=1| 118, 354, 472 }}
@@ Line 31: / Line 34: @@
 Badness: 0.049316
-==== Centenniamajor ====
+=== Peithoian ===
-Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.
+Peithoian is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of peithoian temperament is to expand on that. Named after the minor planet [[wikipedia:118 Peitho|118 Peitho]].
 Subgroup: 2.3.5.7.11
@@ Line 48: / Line 51: @@
 Badness: 0.357
-===== 13-limit =====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 61: / Line 64: @@
 Badness: 0.122
-=== Oganesson ===
+== Oganesson ==
 Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.
@@ Line 76: / Line 79: @@
 [[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068
-{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3442, 3776, 7198cd, 10974bccdd }}
+{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3422, 3776, 7198cd, 10974bccdd }}
 [[Badness]]: 2.66
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 89: / Line 92: @@
 Optimal tuning (CTE): ~1953125/1354752 = 634.0085
-{{Optimal ET sequence|legend=1| 354, 3068e, 3442, 3776, 11682ccdde }}
+{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776, 11682ccdde }}
 Badness: 0.568
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 108: / Line 111: @@
 Badness: 0.172
-==== 17-limit ====
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
@@ Line 121: / Line 124: @@
 {{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}
-Badness: 0.105
+=== 19-limit ===
+The closest superparticular to one step of 118edo is [[171/170]], so 19-limit extension for oganesson is prescribed.
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 4096/4095, 6175/6174, 9801/9800, 14365/14364, 28900/28899, 3438981/3437500
+Mapping: [{{val| 118 0 274 643 1094 499 607 1000}}, {{val| 0 3 0 -5 -11 -1 2 -8}}]
+: mapping generators: ~171/170, ~238/165
+Optimal tuning (CTE): ~238/165 = 634.006
+{{Optimal ET sequence|legend=1| 354, ..., 3422, 3776 }}
+{{Navbox fractional-octave}}
+[[Category:118edo]]